When ax3+bx-6 is divided by x+3,the remainder is 9. Find term of a only . The remainder when 2x3-bx2+2ax-4 when it is divided by x-2

a)  a   = - ( 5 + b ) / 9

b) no remainder

Step-by-step explanation:

A) ( ax^3 + bx -6 ) / ( x + 3 )

Remainder = 9

determine the value of a

we will use the result of  if ( x + a ) divides polynomial g(x) the remainder is g(a)

therefore given that ( x + 3 ) divides f(x) = ax^3 + bx - 6

=  f( -3 ) = 9

= a(- 27 ) - 3b - 6 = 9

= -27a  = 9 + 6 + 3b

therefore the term 'a' = - ( 15 / 27   + 3b / 27 )

                                    = - ( 5/9 + b/9 )

                                 a   = - ( 5 + b ) / 9

b) Find the remainder when (2x^3 - bx^2 + 2ax - 4 ) is divided by (x-2 )

given that ( x -2 ) divides f(x) = 2x^3 - bx^2 + 2ax - 4

also given a =  - ( 5 + b ) / 9  from previous polynomial above

= f(2) = ?

= 2(8) -4b + 4a - 4

= 16 - 4b + 4 ( - ( 5 + b ) / 9 ) - 4

= 16 - 4b + (( -20 - 4b ) / 9) - 4

= 16 - 4b - ( 20 - 4b ) / 9)  - 4

= 16 - ( 32b - 20 ) / 9)  - 4  = ?

therefore the remainder 'b' =  ( -108 + 20 ) / 32  = - 2 3/4

since the remainder is negative there is no remainder then

area of part 1:

a1=base*height/2=(8+2)*(8-2-2-1)/2=10*3/2=15 ft^2

area of part 2:

a2=length*width=(8+2)*1=10 ft2

area of part 3:

a3=length*width=8*2=16 ft2

total shaded area : a1+a2+a3=41 ft2

total area of the reactangle : a=16*8=128 ft2

total area of the nonshaded region : 128-41=87 ft2

the probability is 0